Sobolev regularity for Monge-Amp\`ere type equations

TL;DR

This paper establishes Sobolev regularity results for solutions to Monge-Ampère type equations under certain structural conditions on the cost function and density bounds, extending previous regularity findings.

Contribution

It proves that strictly c-convex potentials in optimal transportation are locally in W^{2,1+κ} under mild conditions, generalizing recent regularity results.

Findings

Solutions belong to W^{2,1+κ}_{loc} for some κ>0

Regularity holds under structural conditions on the cost function

Extends regularity results for Monge-Ampère equations with bounded densities

Abstract

In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to $W^{2,1+\kappa}_{\rm loc}$ for some $\kappa>0$. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Amp\`ere equation with right hand side bounded away from zero and infinity.